Direct minimizing method for Yang-Mills energy over $SO(3)$ bundle

TL;DR

This paper applies the direct minimizing method to find Yang-Mills connections over $SO(3)$ bundles on closed four-manifolds, establishing convergence results and identifying special solutions when convergence fails.

Contribution

It introduces a novel application of the direct minimizing method to Yang-Mills theory on $SO(3)$ bundles, including convergence analysis and the identification of anti-selfdual or selfdual solutions.

Findings

Minimizing sequences converge strongly to a Yang-Mills connection under certain conditions.

If strong convergence fails, an anti-selfdual or selfdual connection is obtained.

The method provides a new approach to studying Yang-Mills connections on four-manifolds.

Abstract

In this paper, we use the direct minimizing method to find Yang-Mills connections for $SO(3)$ bundles over closed four manifolds. By constructing test connections, we prove that a minimizing sequence converges strongly to a minimizer under certain assumptions. In case the strong convergence fails, we find an anti-selfdual (or selfdual) connection.